So, let me get the big thing out of the way right at the front.

(lots of you would probably deny the mere existence of race)

All the current genetic research strongly indicates that race is a not a biologically valid concept, which is vitally important to answering your question. In biological terms, there is no such thing as a discrete "Caucasian" race or a "East Asian" race.

I emphasis that these are not discrete populations because the concept of race, since at least the 19th century and the pseudoscientific definition of race by early anthropologists, has been predicated on there being discrete subgroups within the human species (or even separate species/sub-species) that we call "races". However, extensive research on human genetics has revealed that Homo sapiens is a very genetically homogenous and that there are no discrete groupings with the species.

Now, clearly there are physiological and genetic differences between populations, but the issue is that human genetics are a gradient and so classifying different races is ultimately a completely arbitrary endeavor (from the biological standpoint).

Lewontin in his seminal paper of the genetics of human diversity (that I should stress has largely been born-out by more recent research, some of which I include below) summarizes most succinctly on pg. 385:

Racial classification is an attempt to codify what appear to be obvious nodalities in the distribution of human morphological and cultural traits. The difficulty, however, is that despite the undoubted existence of such nodes in the taxonomic space, populations are sprinkled between the nodes so that boundary lines must be arbitrary. No one would confuse a Papuan aboriginal with any South American Indian, yet no one can give an objective criterion for where a dividing line should be draw in the continuum from South American Indians through Polynesians, Micronesians, Melanesians, to Papuans.

We attach great importance to certain physiological features of human populations in order to classify them (skin color most importantly, but the "East Asian" traits you identify above as well). However, these traits, while being very obvious to visual inspection very often don't correlate with the vast majority of genetic patterning in human populations.

For instance, from the Relethford article, emphasis mine:

These methods are applied to global data on craniometric variation (57 traits) and skin color. Multivariate analysis of craniometric variation shows results similar to those obtained from genetic markers and DNA polymorphisms...The apportionment of genetic diversity in skin color is atypical, and cannot be used for purposes of classification.

In summary, before I get into the meat of answering you question more directly, again from Lewontin, pg. 397. Emphasis mine, but that part (about individuals) is further validated in the much more recent Jorde and Wooding article:

It is clear that our perception of relatively large differences between human races and subgroups, as compared to the variation within these groups, is indeed a biased perception and that, based on randomly chosen genetic differences, human races and populations are remarkably similar to each other, with the largest part by far of human variation being accounted for by the differences between individuals.

Why the long diatribe on race? Well, the fairly extensive research on Central Asian genetics is a perfect example of why the race concept doesn't actually help explain human diversity. A number of genetic studies find that, despite your classification of these populations as appearing largely East Asian, the genetic structure of these Central Asian populations is largely a mixture of "three different genetic components of western Eurasian, eastern Eurasian, and south Asian origin" (Quintana-Murci et al.). Even the Kazakhs you point out in your question follow this broad pattern of "Central Asian mtDNA sequences present[ing] features intermediate between European and eastern Asian sequences" (Comas et al.).

This broadly corresponds with the evidence for the species as a whole, that human biology is clinal in distribution such that you can identify clear differences by looking at extreme poles/nodes (say, Western Europe and East Asia), but - recalling the first Lewontin quote I shared earlier - where to draw a line is entirely arbitrary.

That said, historical events can certainly impact the genetic character of populations and the complexity of Central Asian genetics can be explained broadly by some specific events and processes, but more generally by noting that Central Asia has long been a crossroads for human migration and movement across Eurasia. That the genetics of Central Asia largely grade along an East-West axis belies this use of the region as a crossroads connecting different parts of Eurasia, East to West.

The question then is how far back in time does this admixture go and was there a foundational population in the region (as you suggest) from either Eastern Eurasia or Western Eurasia that may have been later replaced? The picture isn't entirely clear, but it does look like some sort of admixture was occurring as far back as the Paleolithic (ca. 27,000BC-10,000BC), but certainly by the early Neolithic (ca. 10,000BC), long before the changes you note in late antiquity (see Zerjal et al).

Interestingly, it seems that in several cases there is very little instance of genetic change being driven by conquest. For instance, haplogroup 10 and 36 are strong indicators of Altaic-speaking populations. However, as Zerjal et al. note (emphasis mine):

The expanding waves of Altaic-speaking nomads involved not only eastern Central Asia—where their genetic contribution is strong, as is shown in figure 7d—but also regions farther west, like Iran, Iraq, Anatolia, and the Caucasus, as well as Europe, which was reached by both the Huns and the Mongols. In these western regions, however, the genetic contribution is low or undetectable (Wells et al. 2001), even though the power of these invaders was sometimes strong enough to impose a language replacement, as in Turkey and Azerbaijan (Cavalli-Sforza et al. 1994).

In other words, conquest doesn't actually explain the clinal distribution from East to West through Central Asia. To finally answer your question, then, the very strong gradient from East-to-West of genetic material indicates that there was never any strong replacement of a group of people (for example, Indo-European speaking groups by Altaic speaking groups) in genetic terms, but rather a continual admixture of different genetic groupings going all the way back to the Neolithic (and perhaps earlier).

While the distribution of languages and certain phenotypes (e.g. "Caucasian" or "East Asian" traits) certainly may have changed fairly significantly before or during Late Antiquity, this change doesn't necessarily indicate any significant changes to the underlying biology of Central Asian populations. Indeed, the genetic map of Central Asia is highly heterogenous, indicating that genetic changes were historically much stronger at the population level rather than on the regional level.

In summary, there is no master narrative of Central Asian population genetics. Individual populations have highly varying genetic histories and the region as a whole exhibits high admixture of genetic material from across Eurasia, rather than any one set of genetic material (i.e. Altaic populations) replacing a different and earlier set of genetic material (i.e. Indo-European populations). This high level of admixture shouldn't be surprising given that Central Asia has been the crossroads of Eurasia since the Paleolithic and also given that human genetic diversity does not cluster into discrete racial groupings (e.g. "Caucasian" or "East Asian"), but rather is clinally distributed, just as we see in the East-West distribution of genetics in Central Asia. That isn't to say that significant social changes can't have occurred, for instance the well-documented replacement of Indo-European language with Altaic languages, but that these changes do not represent a total replacement of one population by another, but that these replacements are largely social and cultural rather than biological (even though these processes may have contributed to furthering the genetic diversity of Central Asia).

Edit: Typos and grammar.

Sources:

• Comas, David et al. 1998. Trading Genes along the Silk Road: mtDNA Sequences and the Origin of Central Asian Populations. The American Journal of Human Genetics, 63(6):1824–1838.

• Jorde, Lynn B. and Stephen P. Wooding. 2004. Genetic Variation, Classification and 'Race'. Nature Genetics, 36.

• Lewontin, Richard. 1972. The Apportionment of Human Diversity. Evolutionary Biology, 6:381-398.

• Quintana-Murci, Lluis et al. 2004. Where West Meets East: The Complex mtDNA Landscape of the Southwest and Central Asian Corridor. The American Journal of Human Genetics, 74(5):827-845.

• Relethford, John H. 2002. Apportionment of Global Human Genetic Diversity Based on Craniometrics and Skin Color. American Journal of Physical Anthropology, 118(4):393–398.

• Zerjal, Tatiana et al. 2002. A Genetic Landscape Reshaped by Recent Events: Y-Chromosomal Insights into Central Asia. The American Journal of Human Genetics, 71(3):466-482.